I initially ran this through a spreadsheet, found the values of L being 7 and 104, but no others seemed to be likely.

I choose to factorise the equation and shortly found FrankM had related thoughts. My thought was to determine if the difference between two consecutive cubes could be a square in more than two circumstances.

The ratios, column by column, represent the relationship between vertically adjacent values within respective columns. From these ratios it seems that the next "L" is somewhere in the vicinity of 20300. Forget the spreadsheet!!

I am satisfied that somewhere many eons down the number line there are many other "L" values.